The dynamics of two bodies connected by a hinge joint, and moving in a plane under the action of a central gravitational force field is analyzed. Each body is modeled as a rigid massless link with a point mass at one end; their other ends are connected together by a hinge joint. The equations of motion of the connected bodies include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom give rise to a complex dynamical system. Relative equilibria, corresponding to circular orbits of fixed radius, are obtained from these equations of motion. The free dynamics has a symmetry due to the cyclic coordinate representing the true anomaly. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. We carry out stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations from these relative equilibria, to confirm their stability properties. These numerical simulations also suggest the use of shape change to alter the overall orientation and orbit of the assembly.

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.

We develop a discrete maximum principle that yields discrete necessary conditions for optimality. These conditions are in agreement with the usual conditions obtained from the Pontryagin maximum principle and define symplectic algorithms that solve the optimal control problem. We show that our approach allows one to recover most of the classical symplectic algorithms and can be enhanced so that the discrete necessary conditions define symplectic-energy conserving algorithms. Finally we illustrate its use with an example of a sub-Riemannian optimal control problem.

The dynamics and control of spacecraft have been widely studied because of their technological significance. In the classical case the spacecraft is assumed to consist of a single rigid body with three-axis torque inputs and with attitude and rate sensing. In practice, however, the situation may be far more complex. For example, any component of the spacecraft that deforms relative to other components will entail a change in the spacecraft mass distribution; in effect, the spacecraft becomes a multibody system. Similarly, structural flexibility and fuel slosh give rise to vibrational degrees of freedom. Spacecraft control is also exacerbated by sensor and actuator nonlinearities. Traditional actuation devices such as thrusters, reaction wheels, momentum wheels, and control moment gyros entail amplitude and rate saturation constraints, gyroscopic coupling, and coupling between translational and attitude dynamics. Additional difficulties arise when accounting for gravitational effects and external disturbances. All of these issues have technological implications. A fundamental difficulty associated with spacecraft technology is the fact that ground-based testing must occur in a 1-g environment whereas the hardware will operate under zero-g conditions. Consequently, spacecraft control engineering must depend on first-principles analysis as well as extrapolation from 1-g testing. The purpose of this paper is to describe a laboratory-based testbed that will be used to explore various issues and concepts in spacecraft dynamics and control. This testbed is based on a triaxial air bearing to allow experiments involving large-angle, three-axis motion. As a precursor to this testbed, we have also developed an air spindle testbed which allows single-axis rotation

In this paper we derive a discontinuous class of controllers for a Lie algebraic generalization of a class of kinematic nonholonomic systems discussed by Brockett (1988). We use a mix of isospectral and double bracket matrix flows.

R. Brockett [1981] derived a canonical form for a class of controllable systems of the form(x) over dot = B(x)u.He showed later that such systems cannot be stabilized by smooth state feedback. In this paper we analyze a discontinuous stabilizing feedback for a Lie algebraic version of this class of systems.

We use an approach based on sliding mode control to design a feedback which forces the state vector of a nonholonomic integrator to track the desired time function. A discontinuous feedback control is found which allows the /spl epsi/-tracking in spite of the fact that the number of controls is less than the dimensionality of the tracking function. The developed approach is applied to a nonholonomic system which describes the motion of the rotating knife moving on the surface. A numerical example is considered.

Uses an approach based on sliding mode control to design a feedback which stabilizes the origin for a class of nonlinear driftless systems of the form x/spl dot/=B(x)u. introduced by Brockett (1993). Brockett showed that these systems fail his necessary condition for the existence of smooth feedback.< >